IHJPAS. 36(1)2023 

292 
 This work is licensed under a Creative Commons Attribution 4.0 International License 

 

   

 

 

On the Stability and Acceleration of Projection Algorithms 

 

 

 
 

 

Abstract 

     The focus of this paper is the presentation of a new type of mapping called projection Jungck 

zn- Suzuki generalized and also defining new algorithms of various types (one-step and two-step 

algorithms) (projection Jungck-normal 𝒩 algorithm, projection Jungck-Picard algorithm, 
projection Jungck-Krasnoselskii algorithm, and projection Jungck-Thianwan algorithm). The 

convergence of these algorithms has been studied, and it was discovered that they all converge to 

a fixed point. Furthermore, using the previous three conditions for the lemma, we demonstrated 

that the difference between any two sequences is zero. These algorithms' stability was 

demonstrated using projection Jungck Suzuki generalized mapping. In contrast, the rate of 

convergence of these algorithms was demonstrated by contrasting the rates of convergence of the 

various algorithms, leading us to conclude that the projection Jungck-normal 𝒩 algorithm is the 
fastest of all the algorithms mentioned above. 

 

Keywords: metric projection, Jungck-Picard algorithm, Jungck-normal 𝒩 algorithm, Jungck-
Thianwan algorithm, Jungck-Krasnoselskii algorithm, Fixed Point. 

 

1.Introduction and Preliminary 

     There are a lot of published studies that included new algorithms and studied their strong 

convergence and stability. In addition, they proved the rate of convergence of these algorithms, 

see [1-9]. These algorithms are valuable tools used to find the value of the fixed point and to 

resolve some problems. For example, they were used in solving nonlinear differential equations, 

integration problems, etc. 

The algorithms presented by the authors are varied (one-step, two-step, etc.). In 1967[10], scientist 

Jungck introduced a new algorithm called Jungck Picard algorithm, but sometimes it is called 

Jungck algorithm, as it consists of one step 

 

doi,org/10.30526/36.1.2923 

Article history: Received 29 June 2022, Accepted 21 Augest 2022, Published in January 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Zena Hussein Maibed 

Department of Mathematics , College of Education  

for Pure Sciences,Ibn Al –Haitham/ University of 

Baghdad- Iraq. 

mrs_ zena.hussein@yahoo.com 

 
 

Noor Nabil Salem 

Department of Mathematics , College of Education 

for Pure Sciences,Ibn Al –Haitham/ University of 

Baghdad- Iraq. 

Nour.Nabeel1203a@ihcoedu.uobaghdad.edu.iq 

 
 

https://creativecommons.org/licenses/by/4.0/
about:blank
mailto:Nour.Nabeel1203a@ihcoedu.uobaghdad.edu.iq


IHJPAS. 36(1)2023 
 

293 
 

Ψ𝑘𝑛+1 =  𝑇𝑘𝑛 , where 𝑘0 ∈ C, 𝑛 ∈ ℕ. In 2011[11], the author Alfred Olufemi Bosede presented 
an algorithm called  Jungck-krasnoselskii. This algorithm is a special case of Jungck-Mann. The 

Jungck-Krasnoselskii algorithm is defined as follows:    

Ψ𝑛𝑛+1 = ( 1 −  𝛿)Ψ𝑛𝑛 + 𝛿 𝑇𝑛𝑛 , where 𝑛0 ∈ C, 𝑛 ∈ ℕ,  and 𝛿 ∈ (0,1).He also proved the 
stability of Jungck-Mann and Jungck-Krasnoselskii algorithms. On the other hand, V. Brined 

proved in 2004[12] that the Picard algorithm converges faster than the Mann algorithm. In 2008, 

[13] presented a new two-step algorithm named after him. The Thianwan algorithm is defined as 

follows:    

𝓏𝑛+1 = ( 1 −  𝑎𝑛)𝓇𝑛 +  𝑎𝑛 𝑇𝓇𝑛 
    𝓇𝑛 = ( 1 −  𝛽𝑛)𝓏𝑛 +  𝛽𝑛 𝑇𝓏𝑛 , where 𝓏0 ∈ C, 𝑛 ∈ ℕ, 
 and {𝛼𝑛}𝑛=0

∞  and {𝛽𝑛}𝑛=0
∞   are the real sequence in [0,1]. 

In addition, it has been proven the strong and weak convergence of this algorithm in the uniformly 

convex Banach space. Now, we will mention some of the priorities we need:   

Definition (1.1):[12] Let {𝓂𝑛}𝑛=0
∞  , {𝓃𝑛}𝑛=0

∞    are two sequence lies in R such that {𝓂𝑛}𝑛=0
∞      

converge to 𝓂,  {𝓃𝑛}𝑛=0
∞  converge to  𝓃,  and 𝒲 = lim

𝑛→∞

|𝓂𝑛−𝓂|

|𝓃𝑛−𝓃|
 

1. If 𝒲 = 0 ⟶ the sequence {𝓂𝑛}𝑛=0
∞ is converge to 𝓂 faster then {𝓃𝑛}𝑛=0

∞  converge to 𝓃. 
2. If  0 < 𝒲 < ∞  →  {𝓂𝑛 }𝑛=0

∞  and {𝓃𝑛}𝑛=0
∞   have the same rate of convergence. 

Lemma (1.2):[14] Let 𝒰 be a uniformly convex Banach space and {𝓂𝑛}𝑛=0
∞  be any sequence such 

that 0 < 𝔭 ≤ 𝛼𝑛 ≤ 𝔮 < 1, for some 𝔭,𝔮 ∈ ℝ
+ and for all 𝑛 ≥ 1. Let {𝓂𝑛}𝑛=0

∞ and {𝓃𝑛}𝑛=0
∞   are 

two sequences of 𝒰 such that: 
lim

𝑛→∞
sup‖𝓂𝑛‖ ≤ 𝑐,  lim

𝑛→∞
sup‖𝓃𝑛‖ ≤ 𝑐 and  

lim
𝑛→∞

sup‖𝛼𝑛 𝓂𝑛 + (1 − 𝛼𝑛 )𝓃𝑛‖ = 𝑐 for some 𝑐 ≥ 0 Then  

lim
𝑛→∞

‖𝓂𝑛 − 𝓃𝑛‖ = 0  

Definition (1.3): [15] Let Ψ, 𝑇 ∶ 𝐶 →  𝐶 such that 𝑇(𝐶)  Ψ(𝐶) and 𝑝 a coincidence point of Ψ 
and 𝑇, that is, Ψ𝑝 =  𝑇𝑝 =  𝑝. For 𝑎𝑛𝑦 𝒸0 𝐶, let the sequence  {Ψ𝓂𝑛}𝑛=0

∞  generated by the 

algorithm procedure Ψ𝓂𝑛 = 𝑓(𝑇, 𝓂𝑛 )  𝑛 ≥ 0 converge to 𝑝. Let  {Ψ𝓎𝑛 }𝑛=0
∞  ⊂ 𝐶 be an 

arbitrary sequence and set  

𝑛 = 𝑑(Ψ𝓎𝑛+1, 𝑓(𝑇, 𝓎𝑛)) , 𝑛 =  0, 1,· · · . Then, the algorithm  Ψ𝒸𝑛  will be called (Ψ, 𝑇) −

𝑠𝑡𝑎𝑏𝑙𝑒 if and only if lim
𝑛→∞

𝜖𝑛 = 0 implies that lim
𝑛→∞

Ψ𝓎𝑛 = 𝑝.  

Lemma (1.4): [12] If  𝜂  is a real number such that 0 < 𝜂 < 1 and {𝜖𝑛}𝑛=0
∞  is a sequence of 

positive numbers, such that lim
𝑛→∞

𝜖𝑛 = 0 then, for any sequence of positive numbers {𝓂𝑛 }𝑛=0
∞  

satisfying 𝓂𝑛+1 ≤ 𝜂𝓂𝑛 + 𝜖𝑛 
Definition (1.5): [16] the mapping 𝑇: 𝐶 → 𝐶   is said Suzuki if satisfying the following 
condition: 

 
 1

2
‖𝓂 − 𝑇(𝓂)‖ ≤ ‖𝓂 − 𝓃‖ ⟹ ‖𝑇(𝓂) − 𝑇(𝓃)‖ ≤ ‖𝓂 − 𝓃‖, ∀𝓂, 𝓃 ∈ 𝐶  

 

2 .Main Results 

We introduce a new type of mapping called projection Jungck zn-Suzuki generalized and by using 

this type of mapping, we will propose new algorithms and analyse their convergence and rate of 

convergence. 

Definition (2.1): 

Let 𝒳 be a normed space, 𝐶 be a nonempty closed convex subset of  𝒳. A mapping  𝑇, Ψ: 𝐶 →
𝐶  and 𝒫𝒸 are called projection Jungck zn-Suzuki generalized mapping if 
1

2
‖𝑥 − 𝑇(𝑥)‖ ≤ ‖Ψ𝑥 − Ψ𝑦‖ Implies that 

‖𝑇(𝑥) − 𝑇(𝑦)‖ ≤ 𝐿‖Ψ𝑥 − Ψ𝑦‖ + 𝜙
(‖𝑥−𝒫𝒸 (𝑥)‖+‖𝑥−Ψ𝑥‖)

1+𝑚𝑎𝑥 {‖𝒫𝒸 (𝑥)−𝒫𝒸 (𝑦)‖,‖Ψ𝑥−Ψ𝑦‖}
   

𝜙: ℛ+ → ℛ+ is a monotone increasing function such that 𝜙(0) = 0 
and 𝐿 ≤ 1. 



IHJPAS. 36(1)2023 
 

294 
 

 

Definition (2.2): 

The projection Jungck-Picard algorithm is defined as follows:      

Ψ𝑘𝑛+1 = 𝒫𝒸 𝑇𝑘𝑛   , 𝑘0 ∈ C.   
Definition (2.3): 

The projection Jungck-Krasnoselskii is defined as follows:        
Ψ𝑛𝑛+1 = (1 − 𝛿)Ψ𝒫𝒸 (𝑛𝑛) + 𝛿𝒫𝒸 𝑇𝑛𝑛  , 𝑛0  ∈ C   where 𝒫𝒸 is metric projection 
 and  𝛿 ∈ (0,1). 
Definition (2.4): 

The projection Jungck-normal 𝒩 algorithm is defined as follows: 
Ψ𝑢𝑛+1 = 𝒫𝒸 𝑇((1 − 𝛼𝑛 )Ψ𝑢𝑛 + 𝛼𝑛 𝒫𝒸 (𝑢𝑛)) , 𝓊0  ∈ C where 𝛼𝑛 ∈ [0,1] 

and Ψ has property 𝒩, i.e., ΨΨ(𝑥) ≤ Ψ(𝑥), 𝑥 ∈ 𝐶 & Ψ is a linear map  
Definition (2.5): 

The projection Jungck-Thianwan algorithm is defined as follows:   

 Ψ𝓏𝑛+1 = (1 − 𝛼𝑛)Ψ𝒫𝒸 (𝓇𝑛) + 𝛼𝑛𝒫𝒸 𝑇𝓇𝑛 
 Ψ𝓇𝑛 = (1 − 𝛽𝑛)Ψ𝒫𝒸 (𝓏𝑛) + 𝛽𝑛𝒫𝒸 𝑇𝓏𝑛  , 𝓏0 ∈ 𝐶. 
And this mapping is commute if Ψ𝒫𝒸 (𝓍𝑛) = 𝒫𝒸 Ψ(𝓍𝑛). 
Now, we talk about convergence, stability and rate of convergence. 

 

Lemma (2.6): 

Let 𝐶 be a non-empty closed convex subset of a uniformly convex Banach space 𝒰. The mappings 

𝑇, Ψ: 𝐶 → 𝐶 are a projection Jungck zn-Suzuki generalized if  {Ψ𝑢𝑛 } generated by projection 

Jungck-normal 𝒩 algorithm, such that  

1.  lim
𝑛→∞

‖Ψ𝑢𝑛 − 𝑝‖ exists for all 𝑝 ∈ 𝒞ℱ(𝒫𝒸 , 𝑇, Ψ),  where 𝒞ℱ(𝒫𝒸 , 𝑇, Ψ) is the family of a common 

fixed point. 

2.  lim
𝑛→∞

‖ΨΨ𝑢𝑛 − Ψ𝒫𝒸 (𝑢𝑛)‖ = 0 

Proof:   

Let 𝑝 ∈ 𝒞ℱ(𝒫𝒸 , 𝑇, Ψ), 

‖Ψ𝑢𝑛+1 − 𝑝‖ = ‖𝒫𝒸 𝑇((1 − 𝛼𝑛 )Ψ𝑢𝑛 + 𝛼𝑛 𝒫𝒸 (𝑢𝑛)) − 𝑝‖  

≤ ‖𝑇((1 − 𝛼𝑛 )Ψ𝑢𝑛 + 𝛼𝑛 𝒫𝒸 (𝑢𝑛)) − 𝑝‖  

≤ 𝐿‖Ψ((1 − 𝛼𝑛)Ψ𝑢𝑛 + 𝛼𝑛𝒫𝒸 (𝑢𝑛 )) − 𝑝‖ +
𝜙(‖𝑝−𝒫𝒸(𝑝)‖+‖𝑝−Ψ𝑝‖)

1+max{‖𝒫𝒸(𝑝)−𝒫𝒸((1−𝛼𝑛)Ψ𝑢𝑛+𝛼𝑛𝒫𝒸(𝑢𝑛))‖,‖Ψ𝑝−Ψ((1−𝛼𝑛)Ψ𝑢𝑛+𝛼𝑛𝒫𝒸(𝑢𝑛))‖}
                                                           

≤ [(1 − 𝛼𝑛)‖ΨΨ𝑢𝑛 − 𝑝‖ + 𝛼𝑛‖Ψ𝒫𝒸 (𝑢𝑛 ) − 𝑝‖]   

≤ [(1 − 𝛼𝑛)‖Ψ𝑢𝑛 − 𝑝‖ + 𝛼𝑛‖𝒫𝒸 Ψ(𝑢𝑛) − 𝑝‖]  

≤ ‖Ψ𝑢𝑛 − 𝑝‖                       

So, we have  

‖Ψ𝑢𝑛+1 − 𝑝‖ ≤ ‖Ψ𝑢𝑛 − 𝑝‖                                                                  (2.1) 

                        ≤ ‖Ψ𝑢𝑛−1 − 𝑝‖  

                        : 

                        ≤ ‖Ψ𝑢0 − 𝑝‖                                                                                             (2.2) 

From (2.1) and (2.2)  lim
𝑛→∞

‖Ψ𝑢𝑛 − 𝑝‖ is exist 

Now to prove 

lim
𝑛→∞

‖ΨΨ𝑢𝑛 − Ψ𝒫𝒸 (𝑢𝑛)‖ = 0  

Since,  lim
𝑛→∞

‖Ψ𝑢𝑛 − 𝑝‖ = 𝑐  



IHJPAS. 36(1)2023 
 

295 
 

⟹  lim
𝑛→∞

𝑠𝑢𝑝‖Ψ𝑢𝑛 − 𝑝‖ = 𝑐   

Now, 

lim
𝑛→∞

𝑠𝑢𝑝 ‖ΨΨ𝑢𝑛 − 𝑝‖  

≤ lim
𝑛→∞

𝑠𝑢𝑝 ‖Ψ𝑢𝑛 − 𝑝‖ = 𝑐  

So, lim
𝑛→∞

𝑠𝑢𝑝 ‖ΨΨ𝑢𝑛 − 𝑝‖ ≤ 𝑐                                                                                  (2.3)  

To proof  lim
𝑛→∞

𝑠𝑢𝑝 ‖Ψ𝒫𝒸 (𝑢𝑛) − 𝑝‖ ≤ 𝑐  

lim
𝑛→∞

𝑠𝑢𝑝 ‖Ψ𝒫𝒸 (𝑢𝑛) − 𝑝‖  

≤ lim
𝑛→∞

𝑠𝑢𝑝 ‖𝒫𝒸 Ψ(𝑢𝑛) − 𝑝‖  

≤ lim
𝑛→∞

𝑠𝑢𝑝‖Ψ𝑢𝑛 − 𝑝‖ = 𝑐  

So, lim
𝑛→∞

𝑠𝑢𝑝 ‖Ψ𝒫𝒸 (𝑢𝑛) − 𝑝‖ ≤ 𝑐                                                                            (2.4) 

Since 𝑐 = lim
𝑛→∞

𝑠𝑢𝑝‖Ψ𝑢𝑛+1 − 𝑝‖  

= lim
𝑛→∞

𝑠𝑢𝑝‖𝒫𝒸 𝑇((1 − 𝛼𝑛)Ψ𝑢𝑛 + 𝛼𝑛𝒫𝒸 (𝑢𝑛)) − 𝑝‖  

≤ lim
𝑛→∞

𝑠𝑢𝑝‖𝑇((1 − 𝛼𝑛)Ψ𝑢𝑛 + 𝛼𝑛𝒫𝒸 (𝑢𝑛)) − 𝑝‖                                                                                                                              

≤ lim
𝑛→∞

𝑠𝑢𝑝‖(1 − 𝛼𝑛)(ΨΨ𝑢𝑛 − 𝑝) + 𝛼𝑛(Ψ𝒫𝒸 (𝑢𝑛) − 𝑝)‖                                     (2.5) 

≤ lim
𝑛→∞

𝑠𝑢𝑝[(1 − 𝛼𝑛 )‖Ψ𝑢𝑛 − 𝑝‖ + 𝛼𝑛‖𝒫𝒸 Ψ(𝑢𝑛) − 𝑝‖]                                   

≤ lim
𝑛→∞

𝑠𝑢𝑝[(1 − 𝛼𝑛 )‖Ψ𝑢𝑛 − 𝑝‖ + 𝛼𝑛‖Ψ𝑢𝑛 − 𝑝‖]   

= lim
𝑛→∞

𝑠𝑢𝑝‖Ψ𝑢𝑛 − 𝑝‖ = 𝑐  

So, lim
𝑛→∞

𝑠𝑢𝑝‖(1 − 𝛼𝑛)(ΨΨ𝑢𝑛 − 𝑝) + 𝛼𝑛 (Ψ𝒫𝒸 (𝑢𝑛) − 𝑝)‖ = 𝑐 

From (2.3), (2.4), (2.5) and by using lemma (1.2) we get 

lim
𝑛→∞

‖ΨΨ𝑢𝑛 − Ψ𝒫𝒸 (𝑢𝑛)‖ = 0.                                                            

Lemma (2.7): 

Let 𝑇, Ψ: 𝐶 → 𝐶 are a projection Jungck zn-Suzuki generalized if {Ψ𝑘𝑛 } generated by the 

projection Jungck-Picard algorithm, such that  

lim
𝑛→∞

‖Ψ𝑘𝑛 − 𝑝‖ exists for all 𝑝 ∈ 𝒞ℱ(𝒫𝒸 , 𝑇, Ψ)  

Proof: Let, 𝑝 ∈ 𝒞ℱ(𝒫𝒸 , 𝑇, Ψ)   

‖Ψ𝑘𝑛+1 − 𝑝‖ = ‖𝒫𝒸 𝑇𝑘𝑛 − 𝑝‖  

                        ≤ ‖𝑇𝑘𝑛 − 𝑝‖  

                        ≤  𝐿‖Ψ𝑘𝑛 − 𝑝‖ +
𝜙(‖𝑝−𝒫𝒸(𝑝)‖+‖𝑝−Ψ𝑝‖)

1+𝑚𝑎𝑥{‖𝒫𝒸(𝑝)−𝒫𝒸(𝑘𝑛)‖,‖Ψ𝑝−Ψ𝑘𝑛‖}
 

                        ≤ ‖Ψ𝑘𝑛 − 𝑝‖                       

So, we have  

‖Ψ𝑘𝑛+1 − 𝑝‖ ≤ ‖Ψ𝑘𝑛 − 𝑝‖   ⟹ {Ψ𝑘𝑛} is non-increasing sequence                      (2.6) 

                        ≤ ‖Ψ𝑘𝑛−1 − 𝑝‖  

           :   

           ≤ ‖Ψ𝑘0 − 𝑝‖   ⟹   {Ψ𝑘𝑛} is bounded sequence                                 (2.7) 

From (2.6) and (2.7) the lim
𝑛→∞

‖Ψ𝑘𝑛 − 𝑝‖ is exist.   

Lemma (2.8): 

Let 𝑇, Ψ: 𝐶 → 𝐶 be a projection Jungck zn-Suzuki generalized mapping if  {Ψ𝑛𝑛} is generated by 

the projection Jungck-Krasnoselskii algorithm, such that  



IHJPAS. 36(1)2023 
 

296 
 

1. lim
𝑛→∞

‖Ψ𝑛𝑛 − 𝑝‖ exists for all 𝑝 ∈ 𝒞ℱ(𝒫𝒸 , 𝑇, Ψ)  

2. lim
𝑛→∞

‖Ψ𝒫𝒸 (𝑛𝑛) − 𝒫𝒸 𝑇𝑛𝑛 ‖ = 0 

Proof: By following the same steps for the proof of theorem (2.6), we get the wanted results. 

 

 

Lemma (2.9):  

Let 𝑇, Ψ: 𝐶 → 𝐶 are a projection Jungck zn-Suzuki generalized mapping if  {Ψ𝓏𝑛} is generated 

by the projection Jungck-Thianwan algorithm, such that:  

1.  lim
𝑛→∞

‖Ψ𝓏𝑛 − 𝑝‖ exists for all 𝑝 ∈ 𝒞ℱ(𝒫𝒸 , 𝑇, Ψ)  

2.  lim
𝑛→∞

‖Ψ𝒫𝒸 (𝓇𝑛) − 𝒫𝒸 𝑇(𝓇𝑛)‖ = 0   

Proof: 

Proof in the same way as lemma proof (2.6) 

Theorem (2.10): Let  𝑇, Ψ: 𝐶 → 𝐶 are a projection Jungck zn-Suzuki generalized mapping with 

𝐿 ∈ (0,1). Let {Ψ𝑛𝑛} be projection Jungck-Krasnoselskii algorithm converging to  𝑝 where 𝛿 ∈

(0,1). Then, the projection Jungck-Krasnoselskii algorithm is (Ψ, 𝑇, 𝒫𝒸 )-stable. 

Proof: 

Let  {𝓎𝑛} ⊂ 𝐶 and 𝑛 = ‖Ψ𝓎𝑛+1 − 𝑓(𝑇, 𝓎𝑛)‖ 

                                    = ‖Ψ𝓎𝑛+1 − (1 − 𝛿)Ψ𝒫𝒸 (𝓎𝑛) + 𝛿𝒫𝒸 𝑇𝓎𝑛‖ 

So,  𝑛 = ‖Ψ𝓎𝑛+1 − (1 − 𝛿)Ψ𝒫𝒸 (𝓎𝑛) + 𝛿𝒫𝒸 𝑇𝓎𝑛‖ 

If  lim
𝑛→∞

𝑛 = 0 , we get  lim
𝑛→∞

Ψ𝓎𝑛+1 = 𝑝 

Then, the projection Jungck-Krasnoselskii algorithm is (Ψ, 𝑇, 𝒫𝒸 )-stable. 

Theorem (2.11): Let 𝑇, Ψ: 𝐶 → 𝐶 are a projection Jungck zn-Suzuki generalized mapping with 
𝐿 ∈ (0,1). Let {Ψ𝑢𝑛} be a projection Jungck-normal 𝒩 algorithm converging to  𝑝 , where  {𝛼𝑛} 
are sequences in [0,1], such that  0 < 𝛼 ≤ 𝛼𝑛. Then, the projection Jungck-normal 𝒩 algorithm 
is (Ψ, 𝑇, 𝒫𝒸 )-stable. 
Proof: 

Let  {𝓎𝑛} ⊂ 𝐶 and 𝑛 = ‖Ψ𝓎𝑛+1 − 𝑓(𝑇, 𝓎𝑛)‖ 
                                    = ‖Ψ𝓎𝑛+1 − 𝒫𝒸 𝑇((1 − 𝛼𝑛 )Ψ𝓎𝑛 + 𝛼𝑛𝒫𝒸 (𝓎𝑛))‖  

So,  𝑛 = ‖Ψ𝓎𝑛+1 − 𝒫𝒸 𝑇((1 − 𝛼𝑛)Ψ𝓎𝑛 + 𝛼𝑛 𝒫𝒸 (𝓎𝑛))‖ 

If  lim
𝑛→∞

𝑛 = 0, we get  lim
𝑛→∞

Ψ𝓎𝑛+1 = 𝑝 

Then, the projection Jungck-normal 𝒩 algorithm is (Ψ, 𝑇, 𝒫𝒸 )-stable. 
Theorem (2.12): Let 𝑇, Ψ: 𝐶 → 𝐶 are projection Jungck zn-Suzuki generalized mapping with 𝐿 ∈

(0,1). Let {Ψ𝑘𝑛} be a projection Jungck-Picard algorithm converging to 𝑝, where  {𝛼𝑛} are 

sequences in [0,1]. Then, the projection Jungck-Picard algorithm is (Ψ, 𝑇, 𝒫𝒸 )-stable. 

Theorem (2.13): Let 𝑇, Ψ: 𝐶 → 𝐶 are projection Jungck zn-Suzuki generalized mapping with 𝐿 ∈

(0,1). Let {Ψ𝓏𝑛} be a projection Jungck- Thianwan algorithm converging to  𝑝, where  {𝛼𝑛} and 

{𝛽𝑛}  are sequences in [0,1] such that  0 < 𝛼 ≤ 𝛼𝑛  and , 0 < 𝛽 ≤ 𝛽𝑛  then, the projection Jungck-

Thianwan algorithm is (Ψ, 𝑇, 𝒫𝒸 )-stable. 

Proof: By following the same steps of the proof of theorem (2.9), we get the wanted results. 



IHJPAS. 36(1)2023 
 

297 
 

Theorem (2.14): Let  𝑇, Ψ are projection Jungck zn-Suzuki generalized mapping and 
𝒞ℱ(𝒫𝒸 , 𝑇, Ψ)  ≠ 𝜙. Then, the projection Jungck-Picard algorithm converges faster than projection 
Jungck-Krasnoselskii algorithm. 

 

Proof: 

For projection Jungck-Picard algorithm  
‖Ψ𝑘𝑛+1 − 𝑝‖ = ‖𝒫𝒸 𝑇𝑘𝑛 − 𝑝‖  
                ≤ ‖𝑇𝑘𝑛 − 𝑝‖ 

                        ≤ 𝐿‖Ψ𝑘𝑛 − 𝑝‖ +
𝜙(‖𝑝−𝒫𝒸(𝑝)‖+‖𝑝−Ψ𝑝‖)

1+𝑚𝑎𝑥{‖𝒫𝒸(𝑝)−𝒫𝒸(𝑘𝑛)‖,‖Ψ𝑝−Ψ𝑘𝑛‖}
 

                : 

                ≤ 𝐿𝑛 ‖Ψ𝑘0 − 𝑝‖ 
Put  𝒫. 𝒥. 𝑃. 𝒜 = 𝐿𝑛‖Ψ𝑘0 − 𝑝‖ 
For projection Jungck-Krasnoselskii algorithm. 
‖Ψ𝑛𝑛+1 − 𝑝‖ = ‖(1 − 𝛿)Ψ𝒫𝒸 (𝑛𝑛) + 𝛿𝒫𝒸 𝑇𝑛𝑛 − 𝑝‖  
≤ (1 − 𝛿)‖Ψ𝒫𝒸 (𝑛𝑛) − 𝑝‖ + 𝛿‖𝒫𝒸 𝑇𝑛𝑛 − 𝑝‖  
= (1 − 𝛿)‖𝒫𝒸 Ψ(𝑛𝑛) − 𝑝‖ + 𝛿‖𝒫𝒸 𝑇𝑛𝑛 − 𝑝‖  

≤ (1 − 𝛿)‖Ψ𝑛𝑛 − 𝑝‖ + 𝐿𝛿‖Ψ𝑛𝑛 − 𝑝‖ +
𝜙(‖𝑝−𝒫𝒸(𝑝)‖+‖𝑝−Ψ𝑝‖)

1+max{‖𝒫𝒸(𝑝)−𝒫𝒸(𝑛𝑛)‖,‖Ψ𝑝−Ψ𝑛𝑛‖}
  

≤ (1 − 𝛿(1 − 𝐿))‖Ψ𝑛𝑛 − 𝑝‖   
: 

≤ [1 − 𝛿(1 − 𝐿)]𝑛‖Ψ𝑛0 − 𝑝‖  
Put: 

𝒫. 𝒥. 𝒦. 𝒜 = [1 − 𝛿(1 − 𝐿)]𝑛‖Ψ𝑛0 − 𝑝‖  
Now since: 
𝒫.𝒥.𝑃.𝒜

𝒫.𝒥.𝒦.𝒜
=

𝐿𝑛‖Ψ𝑘0−𝑝‖

[1−𝛿(1−𝐿)]𝑛‖Ψ𝑛0−𝑝‖
 → 0   as 𝑛 → ∞.     

Hence, the projection Jungck-Picard algorithm converges to 𝑝 faster than the projection Jungck-
Krasnoselskii algorithm.                                    

Theorem (2.15): Let 𝑇, Ψ be a projection Jungck zn-Suzuki generalized mapping and 

𝒞ℱ(𝒫𝒸 , 𝑇, Ψ)  ≠ 𝜙. Then, the projection Jungck-normal 𝒩 algorithm converges faster than the 

projection Jungck- Picard algorithm. 

Proof: Let 𝑝 ∈ 𝒞ℱ(𝒫𝒸 , 𝑇, Ψ)  and suppose that there exists  ʎ; 0 ≤ ʎ ≤ 𝛼𝑛 ≤ 1 

For projection Jungck –Picard algorithm 

We have, 

  𝒫. 𝒥. 𝑃. 𝒜 = 𝐿𝑛 ‖Ψ𝑘0 − 𝑝‖ 
From projection Jungck-normal 𝒩algorithm 
‖Ψ𝑢𝑛+1 − 𝑝‖ = ‖𝒫𝒸 𝑇((1 − 𝛼𝑛 )Ψ𝑢𝑛 + 𝛼𝑛 𝒫𝒸 (𝑢𝑛)) − 𝑝‖  

≤ ‖𝑇((1 − 𝛼𝑛 )Ψ𝑢𝑛 + 𝛼𝑛 𝒫𝒸 (𝑢𝑛)) − 𝑝‖  

≤ 𝐿‖Ψ((1 − 𝛼𝑛)Ψ𝑢𝑛 + 𝛼𝑛𝒫𝒸 (𝑢𝑛 )) − 𝑝‖ +
𝜙(‖𝑝−𝒫𝒸(𝑝)‖+‖𝑝−Ψ𝑝‖)

1+𝑚𝑎𝑥{‖𝒫𝒸(𝑝)−𝒫𝒸((1−𝛼𝑛)Ψ𝑢𝑛+𝛼𝑛𝒫𝒸(𝑢𝑛))‖,‖Ψ𝑝−Ψ((1−𝛼𝑛)Ψ𝑢𝑛+𝛼𝑛𝒫𝒸(𝑢𝑛))‖}
  

≤ 𝐿‖Ψ((1 − 𝛼𝑛)Ψ𝑢𝑛 + 𝛼𝑛𝒫𝒸 (𝑢𝑛 )) − 𝑝‖  

≤ 𝐿‖(1 − 𝛼𝑛)ΨΨ𝑢𝑛 + 𝛼𝑛Ψ𝒫𝒸 (𝑢𝑛) − (1 − 𝛼𝑛 + 𝛼𝑛)𝑝‖                                                      
≤ 𝐿‖(1 − 𝛼𝑛)(ΨΨ𝑢𝑛 − 𝑝) + 𝛼𝑛 (Ψ𝒫𝒸 (𝑢𝑛) − 𝑝)‖  
≤ 𝐿[(1 − 𝛼𝑛)‖ΨΨ𝑢𝑛 − 𝑝‖ + 𝛼𝑛‖Ψ𝒫𝒸 (𝑢𝑛) − 𝑝‖]   
≤ 𝐿[(1 − 𝛼𝑛)‖Ψ𝑢𝑛 − 𝑝‖ + 𝛼𝑛‖Ψ𝒫𝒸 (𝑢𝑛 ) − 𝑝‖]  
= 𝐿[(1 − 𝛼𝑛)‖Ψ𝑢𝑛 − 𝑝‖ + 𝛼𝑛‖𝒫𝒸 Ψ(𝑢𝑛) − 𝑝‖]  
≤ 𝐿[(1 − 𝛼𝑛)‖Ψ𝑢𝑛 − 𝑝‖ + 𝛼𝑛‖Ψ𝑢𝑛 − 𝑝‖]  
≤ 𝐿[1 − 𝜆(1 − 𝐿)]‖Ψ𝑢𝑛 − 𝑝‖                                           
 : 



IHJPAS. 36(1)2023 
 

298 
 

≤ 𝐿𝑛[1 − 𝜆(1 − 𝐿)]𝑛‖Ψ𝑢0 − 𝑝‖  
Let, 

𝒫. 𝒥. 𝒩. 𝒜 = 𝐿𝑛 [1 − 𝜆(1 − 𝐿)]𝑛‖Ψ𝑢0 − 𝑝‖  
Now, since: 
𝒫.𝒥.𝒩.𝒜

𝒫.𝒥.𝑃.𝒜
=

𝐿𝑛[1−𝛼𝑛(1−𝐿)]
𝑛‖Ψ𝑢0−𝑝‖

𝐿𝑛‖Ψ𝑘0−𝑝‖
 → 0  as 𝑛 → ∞ 

Hence, the projection Jungck-normal 𝒩 algorithm converges to 𝑝 faster than the projection 
Jungck-Picard algorithm.  

      

Theorem (2.16)  

Let 𝒳 be a normed space and C be a nonempty closed convex subset of 𝒳 if  T is a projection 

Jungck zn-Suzuki generalized mapping and 𝒞ℱ(𝒫𝒸 , 𝑇, Ψ) ≠ ϕ. Then, the projection Jungck-

normal 𝒩algorithm converges faster than the projection Jungck-Thianwan algorithm. 

Proof:  Let 𝑝 ∈ 𝒞ℱ(𝒫𝒸 , 𝑇, Ψ)and suppose that there exists  ʎ; 0 ≤ ʎ ≤ 𝛽𝑛, 𝛼𝑛 ≤ 1 

For projection Jungck Thianwan-algorithm  
‖Ψ𝓏𝑛+1 − 𝑝‖ = ‖(1 − 𝛼𝑛)Ψ𝒫𝒸 (𝓇𝑛) + 𝛼𝑛𝒫𝒸 𝑇𝓇𝑛 − (1 − 𝛼𝑛 + 𝛼𝑛)𝑝‖ 
≤ (1 − 𝛼𝑛)‖Ψ𝒫𝒸 (𝓇𝑛) − 𝑝‖ + 𝛼𝑛 ‖𝒫𝒸 𝑇𝓇𝑛 − 𝑝‖ 
= (1 − 𝛼𝑛)‖𝒫𝒸 Ψ(𝓇𝑛) − 𝑝‖ + 𝛼𝑛 ‖𝒫𝒸 𝑇𝓇𝑛 − 𝑝‖ 
≤ (1 − 𝛼𝑛)‖Ψ𝓇𝑛 − 𝑝‖ + 𝛼𝑛 ‖𝑇𝓇𝑛 − 𝑝‖  

≤ (1 − 𝛼𝑛)‖Ψ𝓇𝑛 − 𝑝‖ + 𝐿𝛼𝑛‖Ψ𝓇𝑛 − 𝑝‖ +
𝜙(‖𝑝−𝒫𝒸(𝑝)‖+‖𝑝−Ψ𝑝‖)

1+𝑚𝑎𝑥{‖𝒫𝒸(𝑝)−𝒫𝒸(𝓇𝑛)‖,‖Ψ𝑝−Ψ𝓇𝑛‖}
  

≤ (1 − 𝛼𝑛(1 − 𝐿)‖Ψ𝓇𝑛 − 𝑝‖                                                                           (2.8) 
Now,  
‖Ψ𝓇𝑛 − 𝑝‖ = ‖(1 − 𝛽𝑛)Ψ𝒫𝒸 (𝓏𝑛) + 𝛽𝑛𝒫𝒸 𝑇𝓏𝑛 − (1 − 𝛽𝑛 + 𝛽𝑛)𝑝‖ 
≤ (1 − 𝛽𝑛)‖Ψ𝒫𝒸 (𝓏𝑛) − 𝑝‖ + 𝛽𝑛‖𝒫𝒸 𝑇𝓏𝑛 − 𝑝‖ 
= (1 − 𝛽𝑛)‖𝒫𝒸 Ψ(𝓏𝑛) − 𝑝‖ + 𝛽𝑛‖𝒫𝒸 𝑇𝓏𝑛 − 𝑝‖ 
≤ (1 − 𝛽𝑛)‖Ψ𝓏𝑛 − 𝑝‖ + 𝛽𝑛‖𝑇𝓏𝑛 − 𝑝‖ 

≤ (1 − 𝛽𝑛)‖Ψ𝓏𝑛 − 𝑝‖ + 𝐿𝛽𝑛‖Ψ𝓏𝑛 − 𝑝‖ +
𝜙(‖𝑝−𝒫𝒸(𝑝)‖+‖𝑝−Ψ𝑝‖)

1+𝑚𝑎𝑥{‖𝒫𝒸(𝑝)−𝒫𝒸(𝓏𝑛)‖,‖Ψ𝑝−Ψ𝓏𝑛‖}
  

= (1 − 𝛽𝑛(1 − 𝐿))‖Ψ𝓏𝑛 − 𝑝‖ 

≤ (1 − 𝜆(1 − 𝐿))‖Ψ𝓏𝑛 − 𝑝‖                                                                                     (2.9) 
Substitute Equation (2.9) in to Equation (2.8) 

‖Ψ𝓏𝑛+1 − 𝑝‖ ≤ (1 − 𝛼𝑛 (1 − 𝐿)[(1 − 𝜆(1 − 𝐿))‖Ψ𝓏𝑛 − 𝑝‖]   

= [1 − 𝜆(1 − 𝐿) − 𝛼𝑛 (1 − 𝐿) + 𝛼𝑛𝜆(1 − 𝐿)]‖Ψ𝓏𝑛 − 𝑝‖ 
≤ [1 − 𝜆(1 − 𝐿) − 𝜆(1 − 𝐿) + 𝜆2(1 − 𝐿)]‖Ψ𝓏𝑛 − 𝑝‖ 
≤ [1 − 2𝜆(1 − 𝐿) + 𝜆2(1 − 𝐿)]‖Ψ𝓏𝑛 − 𝑝‖ 
≤ [1 − 𝜆(1 − 𝐿)]2‖Ψ𝓏𝑛 − 𝑝‖ 
: 

≤ [1 − 𝜆(1 − 𝐿)]2𝑛‖Ψ𝓏0 − 𝑝‖ 
Put: 

 𝒫. 𝒥. 𝒯. 𝒜 = [1 − 𝜆(1 − 𝐿)]2𝑛‖Ψ𝓏0 − 𝑝‖ 
From projection Jungck-normal 𝒩algorithm 
𝒫. 𝒥. 𝒩. 𝒜 = 𝐿𝑛 [1 − 𝜆(1 − 𝐿)]𝑛‖Ψ𝑢0 − 𝑝‖  
Now since: 
𝒫.𝒥.𝒩.𝒜

𝒫.𝒥.𝒯.𝒜
=

𝐿𝑛[1−𝜆(1−𝐿)]𝑛‖Ψ𝑢0−𝑝‖

[1−𝜆(1−𝐿)]2𝑛‖Ψ𝓏0−𝑝‖
   as 𝑛 → ∞ 

Hence, the projection Jungck-normal 𝒩 algorithm converges to 𝑝 is faster than the projection 

Jungck- Thianwan algorithm.     

   

 



IHJPAS. 36(1)2023 
 

299 
 

3. Conclusion  

     This paper offered a new type of mapping as well as introduced new algorithms and 

demonstrated their convergence and stability. On the other hand, the acceleration and stability 

were examined. We discovered that the projection Jungck-normal algorithm is quicker than the 

techniques stated in this paper to achieve the fixed point.  

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